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Solving problems can be challenging. If you have to put pen to paper, it's nice to have a process to guide you and aid your thinking. That's why the heuristic shared on this page can be helpful.
Solving problems can be challenging. If you have to put pen to paper, it's nice to have a process to guide you and aid your thinking. That's why the heuristic shared on this page can be helpful.
It was developed by George Polya, a mathematician, and draws its inspiration from Socrates.
It was developed by George Polya, a mathematician, and draws its inspiration from Socrates.
"We all must add the action of your own mind in order to learn something. Socrates expressed it two thousand years ago very colorfully when he said that an idea should be born in the student’s mind, and the teacher should just act as a midwife. The idea should be born in the student’s mind naturally and the midwife shouldn’t interfere too much, too early. But if the labor of birth is too long, the midwife must then intervene. The student learns by his own actions." (Source)
Four Steps to Solving a Problem
Four Steps to Solving a Problem
George Pólya (December 13, 1887 – September 7, 1985) was a Hungarian mathematician.
George Pólya (December 13, 1887 – September 7, 1985) was a Hungarian mathematician.
He was a professor of mathematics from 1914 to 1940 at ETH ZĂĽrich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory.
He was a professor of mathematics from 1914 to 1940 at ETH ZĂĽrich and from 1940 to 1953 at Stanford University. He made fundamental contributions to combinatorics, number theory, numerical analysis and probability theory.
He is also noted for his work in heuristics and mathematics education. He spent considerable effort to identify systematic methods of problem-solving to further discovery and invention in mathematics for students, teachers, and researchers.
He is also noted for his work in heuristics and mathematics education. He spent considerable effort to identify systematic methods of problem-solving to further discovery and invention in mathematics for students, teachers, and researchers.
- Understand the Problem
Read the problem over carefully and ask yourself:
Do I know the meaning of all the words?
What is being asked for?
What is given in the problem?
Is the given information sufficient (for the solution to be unique)?
Is there some inconsistent or superfluous information which is given?
By way of checking your understanding, try restating the problem in a different way.
2. Design a Plan for Solving the Problem
2. Design a Plan for Solving the Problem
Decide how you are going to work on the problem. Try one or more of these strategies:
Draw a picture or diagram. Making a picture which relates the information given to what is asked for can often lead to a solution.
Make a list. This is a strategy which is especially useful in problems where you need to count the members of a set.
Solve smaller versions of the problem and look for a pattern. Can you make problem smaller? Doing so may help you see a pattern to solve the bigger problem.
Decompose the problem. Break problem into a series of smaller problems (or steps).
Use variables and write an equation.
3. Carry Out the Plan
3. Carry Out the Plan
Spend a reasonable amount of time trying to solve the problem using your plan.
If you are not successful, go back to step 2.
If you run out of strategies, go back to step 1.
If you still don't have any luck, talk the problem over with a classmate.
4. Look Back
4. Look Back
After you have a proposed solution, check your solution out.
Is it reasonable?
Is it unique?
Can you see an easier way to solve the problem?
Can you generalize the problem?
A Simple Example
A Simple Example
- Understand the Problem
Read the problem over carefully and ask yourself:
Do I know the meaning of all the words?
What is being asked for?
What is given in the problem?
Is the given information sufficient (for the solution to be unique)?
Is there some inconsistent or superfluous information which is given?
By way of checking your understanding, try restating the problem in a different way.
Problem
Problem
I need to set up online professional development, and be able to track how many people sign up, as well as represent that in graph form. I want participants to get digital certificates upon completion of a course (or submission of a form where they indicate they have reviewed the content).
2. Design a Plan for Solving the Problem
2. Design a Plan for Solving the Problem
Decide how you are going to work on the problem. Try one or more of these strategies:
Draw a picture or diagram. Making a picture which relates the information given to what is asked for can often lead to a solution.
Make a list. This is a strategy which is especially useful in problems where you need to count the members of a set.
Solve smaller versions of the problem and look for a pattern. Can you make problem smaller? Doing so may help you see a pattern to solve the bigger problem.
Decompose the problem. Break problem into a series of smaller problems (or steps).
Use variables and write an equation.
3. Carry Out the Plan
3. Carry Out the Plan
Spend a reasonable amount of time trying to solve the problem using your plan.
If you are not successful, go back to step 2.
If you run out of strategies, go back to step 1.
If you still don't have any luck, talk the problem over with a classmate.
Revisiting and Improving the Planned Solution
Revisiting and Improving the Planned Solution
It's important to implement the plan, try it out, then implement it as needed. Google Sheets made doing that easy since I was able to implement the plan, testing it and refining it along the way.
4. Look Back
4. Look Back
After you have a proposed solution, check your solution out.
Is it reasonable?
Is it unique?
Can you see an easier way to solve the problem?
Can you generalize the problem?
Reflection and Self-Assessment
Reflection and Self-Assessment
Most plans fail in their execution. However, I'm happy to report that this plan worked well. Over twenty thousand people have gone through the process and successfully receive certificates via email in portable document format.
Common Problems and Responses
Common Problems and Responses
Reflection Invitation
Reflection Invitation
Allow me to invite you to reflect on what you are learning today. You can use this Jamboard to take notes on three areas:
What you've seen today
What it means for teaching and learning or your work
And what next steps or changes can you make to your work based on those
You can also get a copy of the Google Slides that serves as the template for the Jamboard here.
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